Subgroups — #38744969

Order 25216.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263269	C₂	1	4	yes	maximal, complemented
2	#27161456	C₂	1	1	yes	maximal, characteristic
3	C₄×#15691333	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₅₇₆	—	2	1	no	—
6	C₂×D₁₅₇₆	C₂²	1	2	yes	—
7	#20263323	—	2	1	no	—
8	#20263269	—	2	8	no	—
9	C₂×#15691333	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₁₉₇	—	2	1	no	—
11	C₂×Q₁₆×C₁₉₇	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₉₇	—	2	1	no	—
13	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
14	C₂×D₇₈₈	—	2	2	no	—
15	C₂×D₇₈₈	—	4	2	no	—
16	D₁₅₇₆	—	2	4	no	—
17	D₁₅₇₆	—	4	2	no	—
18	#15691333	D₄	1	4	yes	complemented
19	C₂×Q₈×C₁₉₇	—	2	2	no	—
20	Q₁₆×C₁₉₇	—	2	4	no	—
21	Q₁₆×C₁₉₇	—	4	2	no	—
22	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₉₇	—	4	2	no	—
25	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
26	C₂²×D₁₉₇	—	4	2	no	—
27	D₇₈₈	—	4	2	no	—
28	D₇₈₈	—	4	2	no	—
29	D₇₈₈	—	8	4	no	—
30	Q₈×C₁₉₇	—	4	2	no	—
31	Q₈×C₁₉₇	—	4	2	no	—
32	C₂×C₄×C₁₉₇	—	4	2	no	—
33	C₂×C₄×C₁₉₇	D₈	1	2	yes	—
34	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₉₇	D₈	1	2	yes	—
36	C₈×C₁₉₇	C₂×D₄	1	2	yes	—
37	C₂²×C₁₉₇	C₂×D₈	1	1	yes	characteristic
38	C₂×D₁₉₇	—	8	2	no	—
39	C₂×D₁₉₇	—	8	4	no	—
40	C₄×C₁₉₇	—	8	2	no	—
41	C₄×C₁₉₇	SD₃₂	1	4	yes	complemented
42	C₄×C₁₉₇	#127	1	1	yes	characteristic
43	C₄×C₁₉₇	C₂×D₈	1	1	yes	characteristic
44	C₂×C₁₉₇	#493	1	1	yes	characteristic
45	C₂×C₁₉₇	C₂×SD₃₂	1	2	yes	—
46	D₁₉₇	—	16	2	no	—
47	C₁₉₇	#2248	1	1	yes	characteristic, complemented, Sylow
48	#2248	—	197	1	no	maximal, Sylow
49	C₂×SD₃₂	—	197	4	no	—
50	#494	D₁₉₇	1	1	yes	characteristic, complemented
51	#493	—	197	1	no	—
52	C₄×C₁₆	—	197	1	no	—
53	C₂×Q₁₆	—	2	1	no	—
54	C₂×Q₁₆	C₂×D₁₉₇	1	2	yes	—
55	C₂×D₈	—	394	1	no	—
56	C₂×D₈	—	197	2	no	—
57	C₄⋊Q₈	—	2	1	no	—
58	#127	—	394	1	no	—
59	SD₃₂	—	394	8	no	—
60	C₂×C₁₆	—	197	2	no	—
61	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
62	C₂×Q₈	—	2	2	no	—
63	C₂×D₄	—	788	2	no	—
64	C₂×D₄	—	394	2	no	—
65	Q₁₆	—	2	4	no	—
66	Q₁₆	—	4	2	no	—
67	D₈	—	788	2	no	—
68	D₈	—	394	4	no	—
69	C₂×C₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
70	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
71	C₄⋊C₄	—	4	2	no	—
72	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
73	C₁₆	—	197	4	no	—
74	C₂³	—	788	2	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	4	2	no	—
77	D₄	—	788	2	no	—
78	D₄	—	788	2	no	—
79	D₄	—	1576	4	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	#15691343	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
83	C₈	#15691343	1	2	yes	—
84	C₈	D₄×D₁₉₇	1	2	yes	—
85	C₂²	—	1576	2	no	—
86	C₂²	—	1576	4	no	—
87	C₂²	C₂×#15691343	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	#20263269	1	4	yes	complemented
90	C₄	#20263322	1	1	yes	characteristic
91	C₄	C₂×#15691343	1	1	yes	characteristic
92	C₂	—	3152	2	no	—
93	C₂	#27161455	1	1	yes	characteristic
94	C₂	C₂×#20263269	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#15691343
commutator, cosocle	C₂×C₈×C₁₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₉₇
Fitting	#494×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×D₈
prime radical	#27161456	C₂
hypercenter	#494	D₁₉₇
hypocenter	C₁₉₇	#2248
LCS	G ▷ C₂×C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#20263269	C₂	semidirect
3. C₄×#15691333	C₂	semidirect
4. #494×C₁₉₇	C₂	semidirect
18. #15691333	D₄	semidirect
41. C₄×C₁₉₇	SD₃₂	semidirect
47. C₁₉₇	#2248	semidirect
50. #494	D₁₉₇	semidirect
89. C₄	#20263269	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#20263269	2
#27161456	2
C₄×#15691333	2
#494×C₁₉₇	2
#2248	197